Remark 2.5 In Section 8 of [1]}, Lusztig proved Theorem REF  for any \(A\in \mathcal {Q}_{{\alpha }}\)  and \(B\in \mathcal {Q}_{{\beta }}\) , which implies that the comultiplication on \(\mathbf {U}^-\)  is a homomorphism of algebras.

